### Abstract

If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a touching point. The main result of this paper is a Crossing Lemma for simple curves: Let X and T stand for the sets of intersection points and touching points, respectively, in a family of n simple curves in the plane, no three of which pass through the same point. If |T|>cn, for some fixed constant c>0, then we prove that |X|=Ω(|T|(loglog(|T|/n))^{1/504}). In particular, if |T|/n→∞ then the number of intersection points is much larger than the number of touching points. As a corollary, we confirm the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between n pairwise intersecting simple closed (i.e., Jordan) curves in the plane, no three of which pass through the same point, is at least (1−o(1))n^{2}.

Original language | English |
---|---|

Pages (from-to) | 908-940 |

Number of pages | 33 |

Journal | Advances in Mathematics |

Volume | 331 |

DOIs | |

Publication status | Published - Jun 20 2018 |

### Fingerprint

### Keywords

- Arrangements of curves
- Combinatorial geometry
- Contact graphs
- Crossing Lemma
- Extremal problems
- Separators

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*331*, 908-940. https://doi.org/10.1016/j.aim.2018.03.015

**A Crossing Lemma for Jordan curves.** / Pach, János; Rubin, Natan; Tardos, G.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 331, pp. 908-940. https://doi.org/10.1016/j.aim.2018.03.015

}

TY - JOUR

T1 - A Crossing Lemma for Jordan curves

AU - Pach, János

AU - Rubin, Natan

AU - Tardos, G.

PY - 2018/6/20

Y1 - 2018/6/20

N2 - If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a touching point. The main result of this paper is a Crossing Lemma for simple curves: Let X and T stand for the sets of intersection points and touching points, respectively, in a family of n simple curves in the plane, no three of which pass through the same point. If |T|>cn, for some fixed constant c>0, then we prove that |X|=Ω(|T|(loglog(|T|/n))1/504). In particular, if |T|/n→∞ then the number of intersection points is much larger than the number of touching points. As a corollary, we confirm the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between n pairwise intersecting simple closed (i.e., Jordan) curves in the plane, no three of which pass through the same point, is at least (1−o(1))n2.

AB - If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a touching point. The main result of this paper is a Crossing Lemma for simple curves: Let X and T stand for the sets of intersection points and touching points, respectively, in a family of n simple curves in the plane, no three of which pass through the same point. If |T|>cn, for some fixed constant c>0, then we prove that |X|=Ω(|T|(loglog(|T|/n))1/504). In particular, if |T|/n→∞ then the number of intersection points is much larger than the number of touching points. As a corollary, we confirm the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between n pairwise intersecting simple closed (i.e., Jordan) curves in the plane, no three of which pass through the same point, is at least (1−o(1))n2.

KW - Arrangements of curves

KW - Combinatorial geometry

KW - Contact graphs

KW - Crossing Lemma

KW - Extremal problems

KW - Separators

UR - http://www.scopus.com/inward/record.url?scp=85047220040&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047220040&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.03.015

DO - 10.1016/j.aim.2018.03.015

M3 - Article

AN - SCOPUS:85047220040

VL - 331

SP - 908

EP - 940

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -